Question

cho x, y thỏa man: x^2 + 6(x+y) +2xy +2y^2+6=0. tìm GTLN , GTNN cua S=x+y

Answer 1

\(\left(x+y\right)^2+6\left(x+y\right)+9+y^2-3=0\)

\(\Leftrightarrow\left(x+y+3\right)^2+y^2-3=0\Leftrightarrow\left(x+y+3\right)^2=3-y^2\le3\)

\(\Rightarrow\left(x+y+3\right)^2\le3\Rightarrow-\sqrt{3}\le x+y+3\le\sqrt{3}\)

\(\Rightarrow-3-\sqrt{3}\le x+y\le-3+\sqrt{3}\)

\(\Rightarrow\left\{{}\begin{matrix}S_{max}=-3+\sqrt{3}\\S_{min}=-3-\sqrt{3}\end{matrix}\right.\)